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Of Protection Current Transformer Under Steady-State Conditions Of OperationDr. Wlodzislaw Kostecki Senior Lecturer PNG University of Technology Department of Electrical and Communication Engineering Lae, Morobe Province, Papua New GuineaCopyright \251 2001 by Wlodzislaw Kostecki All rights reserved

This worksheet demonstrates the use of Maple to describe a current transformer from an over-current protection system.The following Maple techniques are highlighted:Developing the modelAnalyzing the modelPlotting waveforms generated by the current transformer

A through-type protection current transformer (CT) with a rated transformation ratio of 800/5 has an annular ferromagnetic core with a mean diameter of 60 NiMlI21tRw== and an effective cross-sectional area of 100 NiMqJCUjbW1HIiIj. The material of the core has a relative permeability of 5000 for small values of magnetic flux density. The primary consists of a single bus bar passing through the window of the core. A high-voltage ac transmission line current, which flows through the bar primary has a rated frequency of 50 NiMlI0h6Rw==. A load (burden) connected across the secondary terminals of the CT is the operating coil of an overcurrent relay, which controls the trip coil of a circuit breaker. The resistance of the operating coil is 0.05 NiMlJk9tZWdhRw== and inductance 0.15 NiMlI21IRw==. The combined resistance in the secondary is 0.07 NiMlJk9tZWdhRw== and inductance 0.2 NiMlI21IRw==.

\342\200\242 Provide a general description of the CT as a component of the over-current protection system.\342\200\242 Describe the CT in physical and mathematical terms using fundamental laws in their symbolic form.\342\200\242 Develop a mathematically tractable model of the electrical-circuit configuration of the CT.\342\200\242 Produce an equivalent circuit diagram of the CT model including designations of its parameters and terminals.\342\200\242 Obtain symbolic equations describing operation of the CT in terms of the magnetic flux and electrical quantities.\342\200\242 Solve the set of equations to obtain symbolic forms of the functions describing the basic time-varying parameters.\342\200\242 Derive the symbolic functions describing the supplementary time-varying parameters.

Use the numerical values given in Section 1.1 to\342\200\242 obtain plots of waveforms of the CT variables,\342\200\242 obtain numerical forms of the functions describing the time-varying parameters,\342\200\242 compute numerical values of time-varying and time-invariant parameters.\342\200\242 verify Kirchhoff\342\200\231s current law (KCL) at a node of the CT-model equivalent-circuit diagram for an arbitrary value of the independent variable.

\342\200\242 Restore the symbolic forms of the functions describing the time-varying parameters.

A current transformer, known alternatively as a series transformer, is an instrument transformer for the transformation of current. Its primary winding is in series with a conductor of the power system whose current is to be measured or controlled. In through-type CTs, the primary winding is a single-turn winding that is the line conductor and is not an integral part of the CT.The protection CT is a device that acts as a buffer or interface between the power system and the protection gear in the event of a major fault in the system. The essence of a CT is that it should reproduce in its secondary the exact replica of the current waveform present in its primary. This must be ensured over a wide range of magnitudes of the primary current.To specify the highest magnitude, relevant standards use the concept of the rated accuracy limit factor, which is expressed as a multiple of the rated effective value of the primary current. This multiple ranges from 20 to 30 and the CT is required to replicate the primary-current waveform over such a range as faithfully as possible.This implies that the magnetization characteristic, or the B-H curve of the CT-core material must be linear in the entire range. In other words, the core must on no account saturate until the primary current is above the accuracy limit factor. For this reason, the operating point of a CT at the rated primary current is chosen well below the saturation region, typically at B = 0.1 T.If the CT iron-core became prematurely saturated, the CT would no longer reproduce in its secondary the exact replica of the current waveform in its primary. Instead, it would produce a distorted (flattened) version of the fault-current waveform. This translates into a false (lower) magnitude of the secondary current, which \342\200\223 in turn \342\200\223 delays the protection gear to respond to the fault, or it may not operate at all to clear the fault. The likely result is a damage to the system and its components.

To develop a mathematically tractable model of the CT, the following assumptions are made.(1) The secondary winding is uniformly distributed on the CT core and there is no leakage of the magnetic flux from either winding. All the magnetic flux is confined to the core and links all the turns of either winding. In consequence, all the magnetic flux is a mutual flux.(2) The effects of hysteresis and eddy-currents in the CT core are negligible. In consequence, the active power loss in the core is negligible and so is the core-loss current.(3) The magnetization characteristic, B = f(H), or the B-H curve of the CT-core ferromagnetic material is linear for the primary current ranging from its rated effective value up to the value resulting from the given rated accuracy limit factor. In consequence, the relative permeability of the material is constant in this range.(4) All the electrical parameters of either winding and of the burden may be represented as lumped parameters. All these parameters are temperature-independent and time-invariant.(5) The primary-current waveform is an ideal sinusoid.

An equivalent schematic diagram of the CT is instrumental in carrying out its physical and mathematical analyses. It is convenient to produce such a diagram as viewed from the secondary side of the CT. The pertinent diagram is shown in Fig. 1.MFNWtKUb<ob<R=MDLCdNNZwj:tK>H:ToIMOUr:<O`Lo\\jyyyyK>\\J><KB<KF\\KF<LJ<LN\\LN<MR<MV\\MV<NZ<N^\\N^<Ob<Of\\Of<Pj<Pn\\Pn<Qr<Qv\\Qv<R:=R>]R>=SB=SF]SF=TJ=TN]TN=UR=UV]UV=VZ=V^]V^=Wb=Wf]Wf=Xj=Xn]Xn=Yr=yyyI;J:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::rGSvqQL:\\kT`N\\@Nd\\QgqxhXjudT`UkHLJ<LNDhqeYyyyqdYtVeP@QkB@SyWcNAuJfb;YpQieKVh=@sFqvnIa=HfB>ZrX\\k>]<BfIJpAJ<eJ<ljZiMJMXX\\vx=wxpPHlTmYMZhNgtnrlp>dsoML?mpC\\Ls@jpak;EoK<WhXtJiVeLlmpSSmL@HUCIKResIar>ayD`vbIOcdPa`v`IUZusmDTGqp\\XQ>hy^Aq;lQQYLCYPLMLZ]u:`wsxK^dMWIJn@TsEly=mXmoAYvO=VIIlEEjAulwTvr@TStQdYrd=O;AmqYYp@YJas=@W[xS^Is[an[yTpap^DMCEY>pJtLSuLmXeVUiP?DYoEvy=qbMUGIT@uNaekwUy>lLWxkSmlOpUE=NSQLdQloTxtINOLNbEe^wki_ye_[y_ydpx=xhT@cbax\\vedQiva\\OvwRqZ`opQ^ugxZLHob@[>`o]^r@Aimv_svqb>]bp`ITVKS[[VDUeeaBHYsb_fmiR@UrDUCYuECUdEkDY]T\\IScMvoKrT?rPGI^CcbKVfIURkbhaIM=cKccT?bJUVLwFbCF<[wk[WbmxeoWBAgoYdPOv]_E?=yToIUUVgOttshFgsR?e\\iwNqf<Odd=h]OEbgu=]WfArRSiNYFbYiU[C;igNuYIWf?urJucpKH^cWjqEIetPWC_mDfiGAKWqAWSICl;EuIdoSd]SHl]w[kFSUT]yFxqeEstPIw]sFmmvsSY@Wf^=r`ARZgDT[F]]VueTPgRMEIWycTCY>Ob@aYUUSpyvOuU:gRkKVKaCNeX[ERlwbL=DZIUbeCP=c\\as:UxwIRTiIsSf>SdvaHnWdBCXP=h?QF;MsXOh?wxZmd]_BFKGMEB[AhTWF?oFqwvZaT?YtAyHw=DM_UA;EUCcGIBdGhcgxA_TSahrqr]=DgEuMmTKKfY_fgSHPSuwihbAeWwCS]TcGbaSWJaHKkSRqeTmtlwCBUyVQyLaH_aYYISrEePkD]uG`AVEkB<Ecy=Fume:mekQRPghlYTXouPaclkXE?FFmSXIiMoVD]tv?dJscI_DHAv^_hNoEcCSWsGZcc^ubscdbcf^KFsYdQsdoWCX=Y[MFCQhXeD\\ssXkunKbwwhdAtGQFBixa]cgoeIWSsiCiSTv]UGUSeYvfSxWoV^YuCYRHcWgeFOcVmcdlgRSQhrOF^_WymWY[G]YTxwGZGY?oEYwV]EtGkSdMcKCYpADE]r`YX;kI_Sxo]IUaUxCFhgs`myCCCnoVFUV?qXrSfbquSUT`grnoTvgxhSeeKu_=f:iGMqgtIDNUfbQWdCRM=s[WYZkrHQX]OuYuuToereS`yv^guMeYD]TiEhl]BJ=bcGdgeXOCrykymgcNurK[Eh=VSwXFcevgHXUe[SD>OSTsvNaUncfewE]?T]yUtkgI;e<uVw=hEgglyVEUHGGw`Abtwfj[ckiWhceV_s_GEymBKuxxmB;SwbQsIcRLIe\\IG_qIeMhmKU]sxrGFloxIIR<gX_oBsex]=WdUY>iWcKdTAdAcfqGII_hnUwZQY<_wy=UHUc<kVjabSaSTyCRsSpwUZsUYohDUHgGuIsiH_UOUdMcWW[VdqgYkdoURxWGiKUtYYu=hUKyWMrfayJgTqOhAsi]qDmuSlWYbMRtMrJuFfSTMSbnmCdUGQQGBwIIYFKOdF_UYwg]qI\\mG`?FoeWisseKgX[RPuSfCGhiGlSgumv=yy<Iy:SxVUHo;VSWtEYIdcbhkXXwXCKtngwXiw=?EewdRggrWFegDdtprYlo]lAMnrQnJ@sCUoSxn@<OL<OeXxNdyNauTMkW`w>]SHTLqur_EuBuX=@q?QnxuWn@PayU]xRuPOxYMKuSHMQapVSiUNHQktSGtpAIX_<n?lKHhXnQph\\KKqlwDJN`LoPuaxkYyMI<lw<pXXsaLuamk=TRepxtXmAlxvMQT`Y?\\wX<rW@PaxxAeK;mxvMQxiPM=rX<rwpXMpfpae>g^DiqEai<?wE?rXXkRgkG>nCI]rIwg`xSndmvrxYZT`m;VZXVyO>eZNiXPjI@fIgiVYo=ad=VihYZfPa<nxVykiN[`NyfnjB_h=hsffl;VoS>hUNgP`\\eprfwf<GrnwnD?jGO\\E?fI>vp^yrfptWiw_pci^_iqeph^h`Qg^OPtMG_j>mIGrgwstxl]@rrgc<nxfYbM^ZWqyAP_\\NhQnqJIeNPcPHwowcfQ]GwilvoRipGYp=`y]_n:hauA]sVvKhy@gyTyyDp_s_ovobPXcmvfm@pPAaIGeY^sXppgO_iypPxlDIaiI\\OAgUiaNFnMpc__yt`gEa\\Jvt``Z[g]WgrQ^\\xIrfIbrfiR`nrxnnh_Tq]uAk:_xiGlgn^_H`^_\\fO]kaaiO`CFbyafpahM`miaxG_r]iaSAitwwGOb\\>\\`o^R_m\\@_fOa]Gc>>x?@ku_wwOgiGeFwqChpK_gVawrhtZic>hp[oy;haqVn]WakYndwfqqZ:NmEYmMvsdA`cA^uowoXn[Os;oX;VI;BRYJfI:qE:;hA<u=J:<rX:nx:>J<JRY:Vi:Z:>u=ZXq[JYb]YjJQdCx_NatJnt=?odG_HqqTfh=FvyA\\d@^NvuyWvnafcPaEfpuArX>rTnnHO^>hoOicfgqBxuxwjghdJWiyispy_h`wTv\\gial?y_pc^P^x`k\\hnYaqI>e=guYxtUoxBIgpoaPqwcgnaPk[`[m?[jw_tvmeg`IqfhYsbv[f>aChdM^ihX[fIatgZAVyZWnnOmPnopIqf`jAyvdN[sYhd^awws`xqUYhnNyHXlMXvbIpaApeYbQftn`snQxSNer?u]gmiiv_x\\vxxTHgwyps>^iAuWOpfvi:`_dQnvosQYblqcTXgbpanw\\[q`uHaB?jHQuvvcGXvYAl`ifUyjFv[uO_cqr=waivlVWgw`[jqfGVdwnyGwur`oUPy[HfIggnVifqf^veSwbtanrwaGomT^cK>eQnZuxnUhinHntitlhmoqkbYsPVtGAwFoxYAtjost`_WoofI\\xqcPWx]y`]FgYW[Ppe?yqbfhvIymn\\OymeFnNG_vnaQY`ZypM?emosnY^t^`OF]gVhxQdL?l_fq\\yZhV^Uvc?il_fwm>ltqdkq]^AuRf_EVbPwnsHgbOydqs`Hw__jrQtNwqqillWtLqlSvhnG]=ak?qdXikCxhlIq^aj:@dwOxmNepWx=GroqueXniGxox[b_l^hsK?apXx?_oPnwK`mOAyMHtvIfr_ihvxa@s@iv[ObMH]qqg@aevYlqVlro`xid_qrQxkmyuJWoqojyXeyafJ@f<a^yN[AHeLwZwVrQXekqjfIoahgCHu<?]EGd?G`;Ggi@yHf_EfpTWgU_sKhig^vxHg[hxeXhRQ[[ViTqm_ack^fxvlZG]LNbsHZuYfyHy[qiRHoMw^jfde>i:Xedx^xYhdQaVVlkOrBitXgnUH]Oodk`fyFbHNowvrva\\AN]DPZlyd\\hj`?u^wsVIkJgrPWwrXprPpCYtbix^Ibi>ceIa[A]Eamm_pTgot?oII_SH^c@wsgakpyXOSyf<=sOotkqXpeS]kY^KWdIIiCi@yhFoVN_t=?I>]sSyv`ks@;y?kRHAhfiwhWbLGHLId?_t`ISGWh\\GD<mxRwR<SHmSfIcRIgwmqGo?hmwFX]IrMhUqsp]W>uGByWJ=RbSY<EgicWu;fFETFoWkUGNkXN;G;WI^sS`ccxiEDSf>wEpaHXUVUkYoqCEcY@CU>]BUegOscMCxYAG_eNjdkfDP<EKGPOyUOD`kPTrL=KsQjW\\spATktYrhx[TP_`JDPnPpSstkHUXyhPytTl`jrIobMNG]WlXVPDNxhrgunOlRwxwFmwxXMJLkkhq>xy<AP`qjvPPrdTj\\jgyOXijL\\qsUxgLKEXnY]tn`muMph]o[uq[hl\\LlieLfHvBQvF<OxiOFLY_Pj\\djiLVvyVY=wNqY`=O>eTkyWIlOlUrO@liTUyyXQerqdWYuY`xtK]KpDtUQNnipTtYDhqjLvRQXK\\SdeXmpTxtsXDsPmYdHL=ESZeSN=trYVIASyeUSELctPepnmyygAUG`MQYqPmU@<Kmdm>YOhDyq]LfpV:YXj\\PviRKQV:ioIIrQIRWLW;PnlxtVEuPxMYXSJ<VBYne<YJLmVEUgyXGaW;DwlIsb<vOyNdmO:]nJaQ`QsQDn:TY:tp;QyMLM^exEQrAhxbuxNiwe<kReJW@O@QopAV<LN@pXdQXSAYUYLuMXbEVTQJCAlJ]OAprjLqKQRfykf@oCmqPdkV\\YXmY?HVnqnrLn`PUCmVBTNXDOCUXvYk@Alw`UiYUshq]qWKASyqtOAuAepOExAljvLsj<XpQp<HuIQllqWChl?pxT@txQUGiTDuMYlL_eXAQMOUj[lkJlK<xYsLjTXSXLTZQYXETNPjX@MW@xsPsLPPsYX<aTS]l[]s]Tj]`OctofloWHpYdS^hWh`T@\\ufajAqUPMtatSIUO[qnI]xeylTysV]OwxWpakx@LL`t=`u@qlwLOsMU@hXBdPNYVuIJ?IS;lu\\EWS=WY@VIELWXpIxjb\\YgxtTHkrMtauNe]SHyt]trmxOIiutmKm`n?IT`ApQltThQGEXlmx=lN\\UP^UjNAMUhv^AsYuJGq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1 An approximate equivalent circuit diagram of the CT as viewed from its secondary side.

NiMmJSJpRzYjJSRwX3NH \342\200\223 primary current referred to the secondary,NiMmJSJ2RzYjJSRwX3NH \342\200\223 primary-winding voltage drop referred to the secondary (represented as an ac voltage source),NiMmJSJpRzYjJSVlcF9zRw== \342\200\223 primary-winding excitation current referred to the secondary,NiMmJSJpRzYjJSVjcF9zRw== \342\200\223 primary-winding core-loss current referred to the secondary,NiMmJSJpRzYjJSVtcF9zRw== \342\200\223 primary-winding magnetizing current referred to the secondary,NiMmJSJlRzYjJSRwX3NH \342\200\223 primary-winding induced electromotive force referred to the secondary (equal to the electromotive force induced in the secondary winding),NiMmJSJpRzYjJSJzRw== \342\200\223 secondary current,NiMmJSJ2RzYjJSJzRw== \342\200\223 secondary voltage defined as the voltage across the burden,NiMmJSJSRzYjJSVjcF9zRw== \342\200\223 equivalent core-loss resistance referred to the secondary,NiMmJSJMRzYjJSVtcF9zRw== \342\200\223 magnetizing inductance referred to the secondary,NiMmJSJSRzYjJSNlc0c= \342\200\223 equivalent resistance of both windings as viewed from the secondary terminals,NiMmJSJMRzYjJSRsZXNH \342\200\223 equivalent leakage inductance of both windings as viewed from the secondary terminals,NiMmJSJSRzYjJSJiRw== \342\200\223 resistance of the burden,NiMmJSJMRzYjJSJiRw== \342\200\223 inductance of the burden,NiMmJSJaRzYjJSJiRw== \342\200\223 impedance of the burden,k and l \342\200\223 secondary terminals.

N.B. The following analysis uses designations found in the equivalent circuit diagram of the CT of Fig. 1 and other designations that are introduced in the course of the analysis.In analyzing the operation of a CT, the well-established theory of on-load transformer operation is applicable with some specific modifications inherent to the protection CT.restart :The rated transformation ratio of the CT, NiMmJiUiYUc2IyUiaUc2IyUibkc=, is the ratio of the rated primary current, NiMmJSRJX3BHNiMlIm5H, to the rated secondary current, NiMmJSRJX3NHNiMlIm5H, ora[i][n] := I_p[n]/I_s[n] : 'a[i][n]' = a[i][n] ;The turns ratio of the CT, NiMmJSJhRzYjJSJ0Rw==, is the ratio of the number of primary turns, NiMmJSJORzYjJSJwRw==, to the number of secondary turns, NiMmJSJORzYjJSJzRw==, ora[t] := N[p]/N[s] : 'a[t]' = a[t] ;Throughout this worksheet, the turns ratio of the CT is used for convenience.N.B. The turns ratio is equal to the reciprocal of the rated transformation ratio of the CT.A current in the line conductor, NiMtJiUiaUc2IyUiTEc2IyUidEc=, is the primary current, NiMtJiUiaUc2IyUicEc2IyUidEc=, of the CT. The impedance of the primary winding of a through-type CT is negligibly small and does not affect the line current.In a bar-primary, ring-type core CT, the leakage flux of both the primary and secondary is negligible and so are both leakage inductances. The primary resistance is also negligibly small. In consequence, with negligible impedance of the primary, the voltage drop across the primary, NiMtJiUidkc2IyUicEc2IyUidEc=, is very small. Since the primary voltage is small, the core-loss current, NiMtJiUiaUc2IyUjY3BHNiMlInRH, flowing through the primary is also small and may be omitted.The excitation current, NiMtJiUiaUc2IyUjZXBHNiMlInRH, drawn by the CT from the ac voltage source is the sum of the core-loss current, NiMtJiUiaUc2IyUjY3BHNiMlInRH, and the magnetizing current, NiMtJiUiaUc2IyUjbXBHNiMlInRH, ori[ep](t) := i[cp](t) + i[mp](t) : 'i[ep](t)' = i[ep](t) ;With the negligible core-loss current, ori[cp](t) := 0 : 'i[cp](t)' = 0 ;the excitation current becomes practically the magnetizing current, ori[ep](t) := eval(i[ep](t)) : 'i[ep](t)' = i[ep](t) ;The primary-winding magnetizing current produces a magnetomotive force (mmf), NiMtJiUiZkc2IyUjbXBHNiMlInRH, which equals the corresponding current linkage, orf[mp](t) := N[p]*i[mp](t) : 'f[mp](t)' = f[mp](t) ;This mmf produces a co-phasal, mutual magnetic flux, NiMtJiUkcGhpRzYjJSJtRzYjJSJ0Rw==, which is confined to the CT core and links both windings. The flux is given by magnetic Ohm\342\200\231s law, viz.phi[m](t) := f[mp](t)/R[m] : 'phi[m](t)' = phi[m](t) ; phi[m](t) := 'phi[m](t)' :where NiMmJSJSRzYjJSJtRw== is the reluctance of the magnetic core.N.B. A formula for the reluctance is given and briefly discussed later.The formula for NiMtJiUkcGhpRzYjJSJtRzYjJSJ0Rw== yields equation Eq[1] in two unknowns, NiMtJiUkcGhpRzYjJSJtRzYjJSJ0Rw== and NiMtJiUiaUc2IyUjbXBHNiMlInRH:Eq[1] := phi[m](t) - f[mp](t)/R[m] = 0 : Eq[1] ; phi[m](t) := 'phi[m](t)' :The mutual magnetic flux, NiMtJiUkcGhpRzYjJSJtRzYjJSJ0Rw==, induces in the secondary an electromotive force (emf), NiMtJiUiZUc2IyUic0c2IyUidEc=, given by Faraday\342\200\231s law of electromagnetic induction, viz.e[s](t) := N[s]*diff(phi[m](t), t) : 'e[s](t)' = e[s](t) ;This emf circulates a current, NiMtJiUiaUc2IyUic0c2IyUidEc=, in the secondary circuit, which is obtained from Kirchhoff\342\200\231s current law (KCL), viz.i[s](t) := i[p_s](t) - i[mp_s](t) : 'i[s](t)' = i[s](t) ;where NiMtJiUiaUc2IyUkcF9zRzYjJSJ0Rw== and NiMtJiUiaUc2IyUlbXBfc0c2IyUidEc= are the primary and magnetizing current, respectively, referred (reflected) to the secondary and given by the formulaei[p_s](t) := a[t]*i[p](t) : i[mp_s](t) := a[t]*i[mp](t) : 'i[p_s](t)' = i[p_s](t) ; 'i[mp_s](t)' = i[mp_s](t) ; i[p](t) := 'i[p](t)' : i[mp](t) := 'i[mp](t)' : i[s](t) := 'i[s](t)' :The formula obtained from KCL yields equation Eq[2] in two unknowns, NiMtJiUiaUc2IyUjbXBHNiMlInRH and NiMtJiUiaUc2IyUic0c2IyUidEc=:Eq[2] := a[t]*i[p](t) - a[t]*i[mp](t) - i[s](t) = 0 : Eq[2] ;Application of Kirchhoff\342\200\231s voltage law (KVL) to the secondary circuit gives the formulae[s](t) := R[Ts] * i[s](t) + L[Ts] * diff(i[s](t), t) : 'e[s](t)' = e[s](t) ; e[s](t) := 'e[s](t)' :where NiMmJSJSRzYjJSNUc0c= and NiMmJSJMRzYjJSNUc0c= are the total resistance and total inductance of the secondary circuit, respectively.The formula for NiMtJiUiZUc2IyUic0c2IyUidEc= yields equation Eq[3] in two unknowns, NiMtJiUiaUc2IyUjbXBHNiMlInRH and NiMtJiUiaUc2IyUic0c2IyUidEc=:Eq[3] := e[s](t) - R[Ts] * i[s](t) - L[Ts] * diff(i[s](t), t) = 0 : Eq[3] ;Substitution of NiMtJiUiZUc2IyUic0c2IyUidEc= given by Faraday\342\200\231s law of electromagnetic induction yields the desired form of equation Eq[3] in two unknowns, NiMtJiUkcGhpRzYjJSJtRzYjJSJ0Rw== and NiMtJiUiaUc2IyUic0c2IyUidEc=, viz.Eq[3] := subs(e[s](t) = N[s]*diff(phi[m](t), t), lhs(Eq[3])) = 0 : Eq[3] ;

The system of the three equations, Eq[1], Eq[2], and Eq[3], will be solved symbolically for the three unknowns, NiMtJiUiaUc2IyUjbXBHNiMlInRH, NiMtJiUkcGhpRzYjJSJtRzYjJSJ0Rw==, and NiMtJiUiaUc2IyUic0c2IyUidEc= with the primary current, NiMtJiUiaUc2IyUicEc2IyUidEc=, given asi[p](t) := I_p_mx * sin(omega*t + theta[p]) : 'i[p](t)' = i[p](t) ;where:NiMlJ0lfcF9teEc= \342\200\223 amplitude of the primary current,NiMlJm9tZWdhRw== \342\200\223 angular frequency of the system voltage,NiMmJSZ0aGV0YUc2IyUicEc= \342\200\223 initial phase angle, or phase shift of the primary current with respect to the system voltage.N.B. It is convenient to adopt the primary current as a reference waveform with zero initial phase angle. However, all the computations found in this worksheet are valid for a general case when the primary current has an arbitrary non-zero initial phase angle.Execution of the following command line yields the solution of the three simultaneous equations in the form of an unordered solution set.sln := dsolve({Eq[1], Eq[2], Eq[3]}, {i[mp](t), phi[m](t), i[s](t)}) :The formulae for the magnetizing current NiMtJiUiaUc2IyUjbXBHNiMlInRH, mutual magnetic flux NiMtJiUkcGhpRzYjJSJtRzYjJSJ0Rw==, and secondary current NiMtJiUiaUc2IyUic0c2IyUidEc=, extracted from the set of solutions are as follows.assign(sln) :\342\200\242 The magnetizing current flowing through the primary:i[mp](t) := simplify(subs(_C1=0, i[mp](t))) : 'i[mp](t)' = i[mp](t) ;A more transparent form of the above formula isi[mp](t) := collect(collect(i[mp](t), sin(omega*t + theta[p])), cos(omega*t + theta[p])) :'i[mp](t)' = i[mp](t) ;\342\200\242 The mutual magnetic flux in the core:phi[m](t) := simplify(subs(_C1=0, phi[m](t))) : 'phi[m](t)' = phi[m](t) ;A more transparent form of the above formula isphi[m](t) := collect(collect(phi[m](t), sin(omega*t + theta[p])), cos(omega*t + theta[p])) :'phi[m](t)' = phi[m](t) ;\342\200\242 The secondary current:i[s](t) := simplify(subs(_C1=0, i[s](t))) : 'i[s](t)' = i[s](t) ;A more transparent form of the above formula isi[s](t) := collect(collect(i[s](t), sin(omega*t + theta[p])), cos(omega*t + theta[p])) :'i[s](t)' = i[s](t) ;

Using the above solutions, formulae for three supplementary electrical quantities are obtained as follows.\342\200\242 The secondary emf \342\200\223 obtainable from the formulaNiMvLSYlImVHNiMlInNHNiMlInRHKiYmJSJOR0YnIiIiLSUlZGlmZkc2JC0mJSRwaGlHNiMlIm1HRilGKkYue[s](t) := N[s]*diff(phi[m](t), t) : 'e[s](t)' = e[s](t) ;A more transparent form of the above formula ise[s](t) := collect(collect(simplify(e[s](t)), cos(omega*t + theta[p])), sin(omega*t + theta[p])) :'e[s](t)' = e[s](t) ;\342\200\242 The primary emf \342\200\223 obtainable from the formulaNiMvLSYlImVHNiMlInBHNiMlInRHKiYmJSJhR0YpIiIiLSZGJjYjJSJzR0YpRi4=e[p](t) := a[t]*e[s](t) : 'e[p](t)' = e[p](t) ;A more transparent form of the above formula ise[p](t) := collect(collect(simplify(e[p](t)), cos(omega*t + theta[p])), sin(omega*t + theta[p])) :'e[p](t)' = e[p](t) ;\342\200\242 The secondary voltage defined as the voltage across the burden \342\200\223 obtainable from KVLNiMvLSYlInZHNiMlInNHNiMlInRHLCYqJiYlIlJHNiMlImJHIiIiLSYlImlHRidGKUYxRjEqJiYlIkxHRi9GMS0lJWRpZmZHNiRGMkYqRjFGMQ==v[s](t) := R[b] * i[s](t) + L[b] * diff(i[s](t), t) : 'v[s](t)' = v[s](t) ;A more transparent form of the above formula isv[s](t) := collect(collect(simplify(v[s](t)), sin(omega*t + theta[p])), cos(omega*t + theta[p])) :'v[s](t)' = v[s](t) ;

It is convenient to replace the variable NiMqJiUmb21lZ2FHIiIiJSJ0R0Yl with NiMlInhH by puttingomega*t = x ;whencet = x/omega ;Therefore, the relevant formulae obtained earlier in the time domain assume the following forms in the new domain, which may be called the angular domain.i[p](x) := subs(t=x/omega, i[p](t)) : i_p_f := i[p](x) :i[mp](x) := subs(t=x/omega, i[mp](t)) : i_mp_f := i[mp](x) :phi[m](x) := subs(t=x/omega, phi[m](t)) : phi_m_f := phi[m](x) :e[s](x) := subs(t=x/omega, e[s](t)) : e_s_f := e[s](x) :i[s](x) := subs(t=x/omega, i[s](t)) : i_s_f := i[s](x) :e[p](x) := subs(t=x/omega, e[p](t)) : e_p_f := e[p](x) :v[s](x) := subs(t=x/omega, v[s](t)) : v_s_f := v[s](x) :In the above formulae, there are several quantities that must be defined at this point of computations. The respective defining formulae follow.\342\200\242 Amplitude of the primary currentI_p_mx := sqrt(2)*I_p : I_p[mx] = I_p_mx ;where I_p is the effective value of the current defined as the root-mean-square, or rms value.\342\200\242 Reluctance of the CT magnetic circuitR[m] := l[av]/(mu[0]*mu[r]*A) : 'R[m]' = R[m] ;where:NiMmJSJsRzYjJSNhdkc= \342\200\223 mean length of the annular magnetic circuit with mean diameter NiMlImRH, given byl[av] := Pi*d : 'l[av]' = l[av] ;A \342\200\223 active cross-sectional area of the magnetic circuit,NiMmJSNtdUc2IyIiIQ== \342\200\223 absolute permeability given as NiMvJiUjbXVHNiMiIiEqKCIiJSIiIiUjUGlHRiopIiM1JSMtN0dGKg== H/m,NiMmJSNtdUc2IyUickc= \342\200\223 relative permeability of the magnetic-circuit material.Notes:1. An active cross-sectional area of the magnetic core is the net cross-sectional area of the magnetic material. The gross, or geometric cross-sectional area, NiMmJSJBRzYjJSJnRw==, of the core is reduced by the lamination stacking factor, NiMmJSJrRzYjJSJzRw==, which depends on the thickness of the inter-laminar insulation and the tightness of the core clamping. Thus, the active cross-sectional area may be expressed by the formulaNiMvJSJBRyomJiUia0c2IyUic0ciIiImRiQ2IyUiZ0dGKg==with NiMmJSJrRzYjJSJzRw== being typically between 0.9 and 0.95.2. In general, the relative permeability is not constant and depends on the magnetic flux density, B, in the core. If the hysteresis phenomenon in the magnetic material is neglected, the relative permeability may be expressed asNiMvLSYlI211RzYjJSJyRzYjJSJCRyomNyMqJiIiIkYuJkYmNiMiIiEhIiJGLi0lJWRpZmZHNiQtRio2IyUiSEdGOEYuThe adoption of a constant value of NiMmJSNtdUc2IyUickc= is equivalent to the adoption of the average value of the slope of the B-H curve in the range NiM3JCwkJiUiSEc2IyUkbWF4RyEiIkYl, which is usually expressed as followsNiMvJiYlI211RzYjJSJyRzYjJSNhdkcqJiYlIkJHNiMlJG1heEciIiIqJiZGJjYjIiIhRjAmJSJIR0YuRjAhIiI=\342\200\242 Angular frequency of the system voltageomega := 2*Pi*f[n] : 'omega' = omega ;where NiMmJSJmRzYjJSJuRw== is the rated frequency of the system voltage.

N[p] := 1 : N[s]:=160 : f[n] := 50 : I_p := 800 : A := 100*10^(-6) : d :=60*10^(-3) : R[b] := 0.05 : R[Ts] := 0.07 : L[b] := 0.15*10^(-3) : L[Ts] := 0.2*10^(-3) : mu[r] := 5000 : mu[0] := 4*Pi*10^(-7) :IMPORTANT:The next parameter to be entered is the initial phase angle of the primary current. This parameter is critical for the numerical computations to follow. Read first the NOTE before entering its value.NOTE:If the primary current is to be referenced to the system voltage, the initial phase angle NiMmJSZ0aGV0YUc2IyUicEc= will be different than zero. Its value must be given and entered in electrical degrees as\342\200\242 a negative number \342\200\223 for current lagging the voltage,\342\200\242 a positive number \342\200\223 for current leading the voltage.The initial phase angle of the primary current:theta[p] := 0 :theta[p[deg]] := theta[p] :theta[p[rad]] := Pi*theta[p]/180 : theta[p] := theta[p[rad]] :

'N[p]' = N[p] ; 'N[s]' = N[s] ; 'f[n]' = f[n]*Hz ; 'I_p' = I_p*'A' ; 'A' = A*10^6*mm^2 ; 'd' = d*10^3*mm ; 'R[b]' = R[b]*Omega ; 'R[Ts]' = R[Ts]*Omega ; 'L[b]' = evalf(L[b]*10^3, 2)*mH ; 'L[Ts]' = evalf(L[Ts]*10^3, 2)*mH ; 'mu[r]' = mu[r] ; 'mu[0]' = mu[0]*`H/m` ; if abs(frac(theta[p[deg]])) = 0 then print('theta[p]' = theta[p[deg]] * `\260`) else print('theta[p]' = evalf(theta[p[deg]], 3) * `\260`) end if :

According to the common engineering practice, the horizontal-axis scale will be expressed in electrical degrees. The preceding functions represent time-varying quantities in the angular domain, which uses the radian scale. To obtain plots of these functions against the variable NiMlInhH expressed in degrees, this variable must be divided by 180/NiMlI1BpRw==. Thus, a new independent variable is introduced for plotting purposes only, which is labeled NiMlI3huRw== and defined as follows:xn := x/(180/Pi) : 'xn' = xn ;The adoption of electrical degrees for the NiMlInhH-axis scale requires re-defining the functions for plotting purposes as follows:i[p](x) := subs(x=xn, i[p](x)) :i[mp](x) := subs(x=xn, i[mp](x)) :phi[m](x) := subs(x=xn, phi[m](x)) :e[s](x) := subs(x=xn, e[s](x)) :i[s](x) := subs(x=xn, i[s](x)) :e[p](x) := subs(x=xn, e[p](x)) :A procedure for plotting waveforms of the time-varying parameters along with the necessary designations of the axes is as follows.with(plots, display, textplot) :plot_i[p] := plot(i[p](x), x = 0..660, y = -1300..2200, thickness = 2, color = red, labels = [``,``]) :plot_i[mp] := plot(i[mp](x)*10^2, x = 0..660, color = blue) :plot_phi[m] := plot(phi[m](x)*10^7, x = 0..660, color = green) :plot_e[s] := plot(e[s](x)*10^3, x = 0..660, color = cyan) :plot_i[s] := plot(i[s](x)*10^2, x = 0..660, color = brown) :plot_e[p] := plot(e[p](x)*10^5, x = 0..660, color = magenta) :plot_hax := plot(0, x = 0..720, color = black) :ha_s := textplot([648, 200, `x (\260 el.)`], align = RIGHT, font = [HELVETICA, 11]) :s_i := textplot([15, 2100, `i_p (A), i_s (A/100), i_mp (A/100)`], align = RIGHT, font = [HELVETICA, 10]) :s_e := textplot([15, 1800, `e_s (mV), e_p (mV/100)`], align = RIGHT, font = [HELVETICA, 10]):s_phi_s := textplot([15,1500, `f`], align = RIGHT, font = [SYMBOL,12]) :s_phi_u1 := textplot([30, 1490, `_m (`], align = RIGHT, font = [HELVETICA, 10]) :s_phi_u2 := textplot([71, 1490, `m`], align = RIGHT, font = [SYMBOL,11]) :s_phi_u3 := textplot([85, 1490, `Wb/10)`], align = RIGHT, font = [HELVETICA, 10]) :display([ plot_i[p], plot_i[mp], plot_phi[m], plot_e[s], plot_i[s], plot_e[p], plot_hax, ha_s, s_i, s_e, s_phi_s, s_phi_u1, s_phi_u2, s_phi_u3 ], title=`WAVEFORMS OF MAGNETIC FLUX AND ELECTRICAL QUANTITIES`, titlefont=[HELVETICA, BOLD, 12]) ;Fig. 2 Waveforms of magnetic flux and electrical quantities characteristic of the CT, plotted as functions of the variable NiMvJSJ4RyomJSZvbWVnYUciIiIlInRHRic= expressed in electrical degrees:\342\200\242 primary current, i_p, (thick red curve);\342\200\242 magnetizing current flowing through the primary, i_mp, multiplied by NiMqJCIjNSIiIw== (blue curve);\342\200\242 mutual magnetic flux in the core, NiMlJHBoaUc=_m, multiplied by NiMqJCIjNSIiKA== (green curve);\342\200\242 secondary emf, e_s, multiplied by NiMqJCIjNSIiJA== (cyan curve);\342\200\242 secondary current, i_s, multiplied by NiMqJCIjNSIiIw== (brown curve);\342\200\242 primary emf, e_p, multiplied by NiMqJCIjNSIiJg== (magenta curve).Observations from Fig. 2:(1) The primary and secondary currents seem to be co-phasal. In fact, pertinent computation provided later reveals that the secondary current leads the primary current (referred to the secondary) by a very small angle called the phase error of the CT. (Refer also to Fig. 3.)(2) The magnetizing current and magnetic flux are co-phasal.(3) The primary and secondary emfs are co-phasal.(4) The magnetizing current is in quadrature with the primary-induced emf, i.e., lags the emf by 90\272. (Refer also to pertinent computation of the phase difference between the primary NiMlJGVtZkc= and magnetizing current.)(5) The magnetizing current lags the primary current. (Refer also to pertinent computation of the phase difference between the primary current and magnetizing current.)(6) The secondary current lags the secondary-induced emf. (Refer also to pertinent computation of the phase difference between the secondary current and secondary emf.)If parts of waveforms of the primary and secondary current are plotted within a very narrow region where they intersect the horizontal axis, the phase lead of the secondary current can be clearly seen. To this end, a region of length of 2\272 is chosen.To ensure selection of a proper region of the same length irrespective of the initial phase angle of the primary current, a reference abscissa, NiMmJSJ4RzYjJSJyRw==, is defined. It corresponds to the zero-crossing of the primary-current waveform, orx[r] := 180 - signum(theta[p[deg]]) * abs(theta[p[deg]]) :if abs(frac(theta[p[deg]])) = 0 then print('x[r]'=x[r] * `\260`) else print('x[r]'=evalf(x[r], 4) * `\260`) end if :Three more ancillary abscissae are defined for the same purpose, viz.x[b] := x[r] - 1 : x[e] := x[r] + 0.7 : x[hax] := x[r] + 0.99 :The plotting procedure proper follows.plot_i[p] := plot(i[p](x), x = x[b]..x[e], y = -14..24, color = red, labels = [``,``]) :plot_i[s] := plot(i[s](x)*10^2, x = x[b]..x[e], color = blue) :plot_hax := plot(0, x = x[b]..x[hax], color = black) :ha_s := textplot([x[hax] - 0.6, 2, `x (\260 el.)`], align = RIGHT, font = [HELVETICA, 11]) :s_i := textplot([x[b]+0.04, 23, `i_p (A), i_s (A/100)`], align = RIGHT, font = [HELVETICA, 10]) :display([ plot_i[p], plot_i[s], ha_s, s_i ], title=`PARTS OF WAVEFORMS OF PRIMARY AND SECONDARY CURRENT`, titlefont=[HELVETICA, BOLD, 12]);Fig. 3 Parts of waveforms of primary and secondary current to show the phase lead of the secondary current:\342\200\242 primary current, i_p, (red curve);\342\200\242 secondary current, i_s, multiplied by NiMqJCIjNSIiIw== (blue curve).Before proceeding to the next section, all the re-defined functions must be 'forgotten' and their original symbolic forms in the radian angular domain restored.To unassign the new definitions from the names of the functions re-defined above only for plotting purposes, each function name is enclosed in single forward quotes:i[p](x) := 'i[p](x)' : i[mp](x) := 'i[mp](x)' : phi[m](x) := 'phi[m](x)' : e[s](x) := 'e[s](x)' : i[s](x) := 'i[s](x)' : e[p](x) := 'e[p](x)' :

It should be noticed that the time-varying quantities obtained earlier have the following forms in the angular domain:\342\200\242 magnetizing current NiMmJSJpRzYjJSNtcEc=, magnetic flux NiMmJSRwaGlHNiMlIm1H, and secondary current NiMmJSJpRzYjJSJzRw==:NiMvLSYlKXF1YW50aXR5RzYjIiIiNiMlInhHLCYqJiYlImFHRidGKC0lJHNpbkc2IywmRipGKCYlJnRoZXRhRzYjJSJwR0YoRihGKComJkYuNiMiIiNGKC0lJGNvc0dGMUYoRig=\342\200\242 secondary emf NiMmJSJlRzYjJSJzRw== and secondary voltage NiMmJSJ2RzYjJSJzRw==:NiMvLSYlKXF1YW50aXR5RzYjIiIjNiMlInhHLCYqJiYlImFHNiMiIiJGMC0lJGNvc0c2IywmRipGMCYlJnRoZXRhRzYjJSJwR0YwRjBGMComJkYuRidGMC0lJHNpbkdGM0YwRjA=Either form may be transformed to an equivalent single-trigonometric-function form, which is more suitable for the analysis of such quantities. The equivalent forms are, respectively,NiMvLSYlKXF1YW50aXR5RzYjIiIiNiMlInhHKiYlIkFHRigtJSRzaW5HNiMsKEYqRigmJSZ0aGV0YUc2IyUicEdGKCYlJmFscGhhR0YnRihGKA==NiMvLSYlKXF1YW50aXR5RzYjIiIjNiMlInhHKiYlIkFHIiIiLSUkc2luRzYjLChGKkYtJiUmdGhldGFHNiMlInBHRi0mJSZhbHBoYUdGJ0YtRi0=where the function parameters denote:NiMlIkFH \342\200\223 amplitude, which is determined from the formulaNiMvJSJBRy0lJXNxcnRHNiMsJiokJiUiYUc2IyIiIiIiI0YtKiQmRis2I0YuRi5GLQ==NiMmJSZhbHBoYUc2IyIiIg==, NiMmJSZhbHBoYUc2IyIiIw== \342\200\223 initial phase angles in radians, which are determined from the formulaeNiMvJiUmYWxwaGFHNiMiIiIqJi0lJXNpZ25HNiMmJSJhRzYjIiIjRictJSdhcmN0YW5HNiMtJSRhYnNHNiMqJkYsRicmRi1GJiEiIkYnNiMvJiUmYWxwaGFHNiMiIiMqJi0lJXNpZ25HNiMmJSJhR0YmIiIiLSUnYXJjdGFuRzYjLSUkYWJzRzYjKiYmRi02I0YuRi5GLCEiIkYuAll the time-varying parameters of the CT will be computed by numerical evaluation of their respective symbolic formulae. Where appropriate, they will be transformed to the form containing the sin function. The transformed form will be given with the phase angle in electrical radians and degrees. The latter form will be given with rounded-off amplitude and phase angle, and with a practical unit of measure, according to the commonly adopted engineering practice.N.B. In all the formulae that follow, the variable x is in radians.

N.B. With negative values of the initial phase angle, NiMmJSZ0aGV0YUc2IyUicEc=, of the primary current, the latter is returned in the form containing the cos function. This form poses a slight inconvenience in the further analysis. It is, therefore, transformed into an equivalent form containing the sin function. A programming construct if ... then ... end if is used to control the final form of the equation for the primary current.i[p_v](x) := evalf(eval(i_p_f)) :if signum(theta[p[deg]]) = -1 then i[p_v](x) := -op(1, i[p_v](x)) * sin(x + evalf(theta[p])) end if :In effect, irrespective of the value (negative, zero, or positive) of the initial phase angle, the primary current is input and displayed with the sinusoidal term, viz.i[p](x) = i[p_v](x) * 'A' ;I_p_mx := op(1, i[p_v](x)) :theta[i_p] := evalf(theta[p]) :theta[deg][i_p] := evalf(180*theta[i_p]/Pi) : theta[deg][i_p_r] := evalf(theta[deg][i_p], 3) :i[p](x) = evalf(I_p_mx, 5) * sin(180*x/Pi + theta[deg][i_p_r] * `\260`) * 'A' ;

i[p_s](x) = 'a[t]' * i[p](x) ; i[p_s_v](x) := a[t]*i[p_v](x) : i[p_s](x) = i[p_s_v](x) * 'A' ;I_p_s_mx := evalf(a[t]*I_p_mx) :theta[deg][i_p_s] := theta[deg][i_p] : theta[deg][i_p_s_r] := evalf(theta[deg][i_p_s], 3) :i[p_s](x) = evalf(I_p_s_mx, 3) * sin(180*x/Pi + theta[deg][i_p_s_r] * `\260`) * 'A' ;The effective value of the primary current referred to the secondary:I_p_s := evalf(I_p_s_mx/sqrt(2)) : 'I_p_s' = evalf(I_p_s, 2) * 'A' ;

\342\200\242 original form:i[mp_v](x) := evalf(eval(i_mp_f)) : i[mp](x) = i[mp_v](x) ;\342\200\242 transformed forms:a[1] := op(1, op(1, i[mp_v](x))) : a[2] := op(1, op(2, i[mp_v](x))) :I_mp_mx := sqrt(a[1]^2 + a[2]^2) :alpha[1] := sign(a[2]) * evalf(arctan(abs(a[2]/a[1]))) :theta[i_mp] := evalf(theta[p] + alpha[1]) :theta[deg][i_mp] := evalf(180*theta[i_mp]/Pi) : theta[deg][i_mp_r] := evalf(theta[deg][i_mp], 3) :i[mp_v](x) := I_mp_mx*sin(x + theta[i_mp]) : i[mp](x) = i[mp_v](x) * 'A' ;i[mp](x) = evalf(I_mp_mx, 3) * sin(180*x/Pi + theta[deg][i_mp_r] * `\260`) * 'A' ;The angular displacement of the magnetizing current with respect to the primary current:delta[1] := theta[deg][i_mp] - theta[deg][i_p] :'theta[deg][i_mp] - theta[deg][i_p]' = evalf(delta[1], 3) * `\260` ;N.B. This result implies that the magnetizing current lags the primary current.The actual effective value of the magnetizing current flowing through the primary:I_mp := evalf(I_mp_mx/sqrt(2)) : 'I_mp' = evalf(I_mp, 2) * 'A' ;The percentage current error of the CT at the adopted magnitude of the primary current, defined as the effective value of the magnetizing current expressed as a percentage of the primary current:epsilon[i] = '100*I_mp/I_p' ; epsilon[i] := 100*I_mp/I_p : epsilon[i] := evalf(epsilon[i], 2) : 'epsilon[i]' = epsilon[i] * convert(convert(convert([37], 'bytes'), string), symbol) ;N.B. As expected and desired, the percentage current error of the CT at the adopted magnitude of the primary current is very low.The effective value of the magnetic field strength in the core:H = 'I_mp*N[p]/l[av]' ; H := I_mp*N[p]/l[av] : 'H' = evalf(H, 3) * `A/m` ; H_v := H :

i[mp_s](x) = 'a[t]' * i[mp](x) ; i[mp_s_v](x) := a[t]*i[mp_v](x) : i[mp_s](x) = i[mp_s_v](x) * 'A' ;I_mp_s_mx := evalf(a[t]*I_mp_mx) :theta[deg][i_mp_s] := theta[deg][i_mp] : theta[deg][i_mp_s_r] := evalf(theta[deg][i_mp_s], 3) :i[mp_s](x) = evalf(I_mp_s_mx*10^3, 3) * sin(180*x/Pi + theta[deg][i_mp_s_r] * `\260`) * mA ;The effective value of the magnetizing current referred to the secondary:I_mp_s := evalf(I_mp_s_mx/sqrt(2)) : 'I_mp_s' = evalf(I_mp_s*10^3, 3) * mA ;

\342\200\242 original form:phi[m_v](x) := evalf(eval(phi_m_f)) : phi[m](x) = phi[m_v](x) ;\342\200\242 transformed forms:a[1] := op(1, op(1, phi[m_v](x))) : a[2] := op(1, op(2, phi[m_v](x))) :Phi_m_mx := sqrt(a[1]^2 + a[2]^2) :alpha[1] := sign(a[2]) * evalf(arctan(abs(a[2]/a[1]))) :theta[phi_m] := evalf(theta[p] + alpha[1]) : theta[deg][phi_m] := evalf(180*theta[phi_m]/Pi) :theta[deg][phi_m_r] := evalf(theta[deg][phi_m], 3) :phi[m_v](x) := Phi_m_mx*sin(x + theta[phi_m]) : phi[m](x) = phi[m_v](x) * Wb ;phi[m](x) = evalf(Phi_m_mx*10^6, 3) * sin(180*x/Pi + theta[deg][phi_m_r] * `\260`) * mu*Wb;The maximum, or peak value of the magnetic flux in the core:Phi_m_mx := Phi_m_mx : Phi[m][mx] = evalf(Phi_m_mx * 10^6, 3) * mu*Wb ;The effective value of the magnetic flux in the core:Phi[m] := evalf(Phi_m_mx/sqrt(2)) : 'Phi[m]' = evalf(Phi[m] * 10^6, 2) * mu*Wb ;The peak value of the magnetic flux density in the core:B[mx] = 'Phi[m][mx]/A' ; B[mx] := Phi_m_mx/A : 'B[mx]' = evalf(B[mx], 2) * T ;N.B. The peak value of the magnetic flux density is required if the hysteresis loss is to be calculated.The effective value of the magnetic flux density in the core:B := evalf(B[mx]/sqrt(2)) : 'B' = evalf(B, 2) * T ;N.B. The effective value of the magnetic flux density is required if the eddy-current loss is to be calculated.N.B. Recall that the values of the magnetic field strength and flux density correspond to the operation of the CT at the rated primary current. If a prospective short-circuit current is k times greater than the rated primary current, both the corresponding magnetic field strength and flux density will also be k times greater at a constant relative permeability, NiMmJSNtdUc2IyUickc=, of the CT-core material. Exemplarily, if the multiple k isk := 25 : 'k' = k ;then the magnetic field strength and the flux density at short-circuit would be, respectively'H[sc]' = 'k * H' ; H[sc] := k*H : H[sc_r] := evalf(H[sc], 4) : 'H[sc]' = H[sc_r] * `A/m` ;'B[sc]' = 'k * B' ; B[sc] := k*B : B[sc_r] := evalf(B[sc], 3) : 'B[sc]' = B[sc_r] * T ;Comment: With the assumed value of the multiple k, magnitudes of the primary current vary in the rangerange_I_p = '[ - k * I_p, k * I_p ]' ; if frac(k*I_p/10^3) = 0 then print(range_I_p = [round(-k*I_p/10^3), round(k*I_p/10^3)] * `kA`) else print(range_I_p = evalf([-k*I_p/10^3, k*I_p/10^3], 3) * `kA`) fi :This implies that the values of the magnetic field strength and magnetic flux density will vary in the following ranges, respectivelyrange_H = '[ -H[sc], H[sc] ]' ; range_H = [ -H[sc_r], H[sc_r] ] * `A/m` ;range_B = '[ -B[sc], B[sc] ]' ; range_B = [ -B[sc_r], B[sc_r] ] * T ;This calls for B to be a linear function of H for a given material at least within the above-determined range of magnetic field strength.The angular displacement of the magnetic flux with respect to the magnetizing current:delta[2] := theta[deg][phi_m] - theta[deg][i_mp] :theta[deg][phi*_m] - 'theta[deg][i_mp]' = round(delta[2]) * `\260`;N.B. The above zero angular displacement verifies that the magnetic flux is co-phasal with the magnetizing current.

\342\200\242 original form:e[s_v](x) := evalf(eval(e_s_f)) : e[s](x) = e[s_v](x) ;\342\200\242 transformed forms:a[1] := op(1, op(2, e[s_v](x))) : a[2] := op(1, op(1, e[s_v](x))) :E_s_mx := sqrt(a[1]^2 + a[2]^2) :alpha[2] := sign(a[2]) * evalf(arctan(abs(a[2]/a[1]))) :theta[e_s] := evalf(theta[p] + alpha[2]) :theta[deg][e_s] := evalf(180*theta[e_s]/Pi) : theta[deg][e_s_r] := evalf(theta[deg][e_s], 3) :e[s_v](x) := E_s_mx*sin(x + theta[e_s]) : e[s](x) = e[s_v](x) * V ;e[s](x) = evalf(E_s_mx, 2) * sin(180*x/Pi + theta[deg][e_s_r] * `\260`) * V ;The effective value of the secondary emf:(a) computed from the maximum value of the secondary emf:E[s] := evalf(E_s_mx/sqrt(2)) : 'E[s]' = evalf(E[s], 2) * V ;(b) computed from the transformer effective emf equation for the secondary:'E[s]' = 'omega*N[s]*Phi[m][mx]/sqrt(2)' ; E[s] := evalf(omega*N[s]*Phi_m_mx/sqrt(2)) : 'E[s]' = evalf(E[s], 2) * V ;

\342\200\242 original form:i[s_v](x) := evalf(eval(i_s_f)) : i[s](x) = i[s_v](x) ;\342\200\242 transformed forms:a[1] := op(1, op(1, i[s_v](x))) : a[2] := op(1, op(2, i[s_v](x))) :I_s_mx := sqrt(a[1]^2 + a[2]^2) :alpha[1] := sign(a[2]) * evalf(arctan(abs(a[2]/a[1]))) :theta[i_s] := evalf(theta[p] + alpha[1]) : theta[deg][i_s] := evalf(180*theta[i_s]/Pi) :if theta[p] = 0 then rd := 2 else rd := 4 end if :theta[deg][i_s_r] := evalf(theta[deg][i_s], rd) :i[s_v](x) := I_s_mx*sin(x + theta[i_s]) : i[s](x) = i[s_v](x) * 'A' ;i[s](x) = evalf(I_s_mx, 3) * sin(180*x/Pi + theta[deg][i_s_r] * `\260`) * 'A' ;The phase error of the CT at the adopted magnitude of the primary current, defined as the angular displacement between the secondary current and the primary current referred to the secondary:epsilon[theta] = abs(abs('theta[deg][i_s]') - abs('theta[deg][i_p_s]')) ; epsilon[theta] := abs(abs(theta[deg][i_s]) - abs(theta[deg][i_p_s])) : epsilon[theta] := evalf(epsilon[theta], 2) : 'epsilon[theta]' = epsilon[theta] * `\260`;N.B. As expected and desired, the phase error of the CT is a small angle, by which the secondary current leads the primary current referred to the secondary.The angular displacement of the secondary current with respect to the secondary emf:delta[3] := theta[deg][i_s] - theta[deg][e_s] :'theta[deg][i_s] - theta[deg][e_s]' = evalf(delta[3], 3) * `\260` ;N.B. This result verifies that the secondary current lags the secondary emf.The effective value of the secondary current:I_s := evalf(I_s_mx/sqrt(2)) : 'I_s' = evalf(I_s, 3) * 'A' ;N.B. Disregarding the magnetizing current flowing through the primary, the product of the turns ratio with the effective value of the primary current should be approximately equal to the effective value of the secondary current computed above. Thus,'a[t] * I_p' = 'I_s' ; 'a[t] * I_p' = round(a[t]*I_p) * 'A' ;

e[p](x) = 'a[t]*e[s](x)' ; e[p_v](x) := a[t]*e[s_v](x) : e[p](x) = e[p_v](x) * V ;E_p_mx := evalf(a[t]*E_s_mx) :theta[deg][e_p] := theta[deg][e_s] : theta[deg][e_p_r] := evalf(theta[deg][e_p], 3) :e[p](x) = evalf(E_p_mx*10^3, 3) * sin(180*x/Pi + theta[deg][e_p_r] * `\260`) * mV ;The angular displacement of the magnetizing current with respect to the primary emf:delta[4] := theta[deg][i_mp] - theta[deg][e_p] :'theta[deg][i_mp]' - 'theta[deg][e_p]' = evalf(delta[4], 3) * `\260`;N.B. The magnetizing current is, as expected, in quadrature with the primary emf, i.e., lags the emf by 90\260.The effective value of the primary emf:(a) computed from the maximum value of the primary emf:E[p] := evalf(E_p_mx/sqrt(2)) : 'E[p]' = evalf(E[p]*10^3, 3) * mV ;(b) computed from the transformer effective emf equation for the primary:'E[p]' = 'omega*N[p]*Phi[m][mx]/sqrt(2)' ; E[p] := evalf(omega*N[p]* Phi_m_mx/sqrt(2)) : 'E[p]' = evalf(E[p]*10^3, 3) * mV ;

\342\200\242 original form:v[s_v](x) := evalf(eval(v_s_f)) : 'v[s](x)' = v[s_v](x) ;\342\200\242 transformed forms:a[1] := op(1, op(2, v[s_v](x))) : a[2] := op(1, op(1, v[s_v](x))) :V_s_mx := sqrt(a[1]^2 + a[2]^2) :alpha[2] := sign(a[2]) * evalf(arctan(abs(a[2]/a[1]))) :theta[v_s] := evalf(theta[p] + alpha[2]) :theta[deg][v_s] := evalf(180*theta[v_s]/Pi) : theta[deg][v_s_r] := evalf(theta[deg][v_s], 3) :v[s_v](x) := V_s_mx*sin(x + theta[v_s]) : 'v[s](x)' = v[s_v](x) * V ;'v[s](x)' = evalf(V_s_mx, 2) * sin(180*x/Pi + theta[deg][v_s_r] * `\260`) * V ;The angular displacement of the secondary current with respect to the secondary voltage:delta[5] := theta[deg][i_s] - theta[deg][v_s] :'theta[deg][i_s] - theta[deg][v_s]' = evalf(delta[5], 3) * `\260` ;N.B. This result verifies that the secondary current lags the secondary voltage. The obtained value of the phase difference is close to a typical value of \342\200\22345\302\272.The effective value of the secondary voltage:V[s] := evalf(V_s_mx/sqrt(2)) : 'V[s]' = evalf(V[s], 2) * V ;Verification of KCL at a node of the equivalent circuit diagram of the CT modelKirchhoff\342\200\231s current law applied to a selected node has the following form in the angular domain:KCL := i[p_s](x) - i[s](x) - i[mp_s](x) = 0 : KCL ;Substitution of transformed forms of the currents results in the following form of KCL:KCL:=subs(i[p_s](x) = i[p_s_v](x), i[s](x) = i[s_v](x), i[mp_s](x) = i[mp_s_v](x), KCL) : KCL ;Choosing x = 60\272 and substituting its radian equivalent in the left-hand side of the KCL equation givelhs_KCL := subs(x = 60*Pi/180, lhs(KCL)) : 'lhs_KCL' = lhs_KCL ;Floating-point evaluation of the above result yieldslhs_KCL := evalf(lhs_KCL) : 'lhs_KCL' = lhs_KCL ;Since the deviation from the expected zero is practically negligible, verification of KCL is considered satisfactory.

'R[m]' = 'l[av]/(mu[0]*mu[r]*A)' ; R[m] := R[m] : 'R[m]' = round(R[m]) * `1/H` ;

L[mp] = 'N[p]^2/R[m]' ; L[mp]:=N[p]^2/R[m] : 'L[mp]' = evalf(L[mp]*10^6, 3) * mu*'H' ;

X[mp] = 'omega*L[mp]' ; X[mp] := omega*L[mp] : 'X[mp]' = evalf(X[mp]*10^3, 3) * m*Omega ;

L[mp_s] = 'L[mp]/a[t]^2' ; L[mp_s_v] := L[mp]/a[t]^2 : L[mp_s] = evalf(L[mp_s_v]*10^3, 3) * mH ;

X[mp_s] = 'X[mp]/a[t]^2' ; X[mp_s_v] := X[mp]/a[t]^2 : X[mp_s] = evalf(X[mp_s_v], 3)*Omega ;

'F[m]' = 'N[p]*I_mp' ; F[m] := N[p]*I_mp : 'F[m]' = evalf(F[m], 3) * 'A' ;

'F[m]' = H * 'l[av]' ; H := H_v : F[m] := evalf(H*l[av]) : 'F[m]' = evalf(F[m], 3) * 'A' ;

'F[m]' = 'Phi[m]*R[m]' ; F[m] := evalf(Phi[m]*R[m]) : 'F[m]' = evalf(F[m], 3) * 'A' ;

Z[Ts] := evalf(sqrt(R[Ts]^2 + (omega*L[Ts])^2)) : 'Z[Ts]' = evalf(Z[Ts]*10^3, 3) * m*Omega ;

ratio[Xmp/Z] := X[mp_s_v]/Z[Ts] : 'X[mp_s]/Z[Ts]' = evalf(ratio[Xmp/Z], 4) ;N.B. This ratio is, as expected, equal to the reciprocal of the ratio of effective values of the primary-winding magnetizing current referred to the secondary and the secondary current, i.e.,reciprocal_ratio[Imp_s/Is] := 1/(I_mp_s/I_s) :'1/(I_mp_s/I_s)' = evalf(reciprocal_ratio[Imp_s/Is], 4) ;N.B. Since the total impedance of the secondary circuit, NiMmJSJaRzYjJSNUc0c=, is very small compared with the magnetizing reactance referred to the secondary, NiMmJSJYRzYjJSVtcF9zRw==, or'Z[Ts]' = evalf('X[mp_s]'/ratio[Xmp/Z], 2) ;almost all the primary current referred to the secondary passes through the burden and the CT operates similarly as a voltage transformer on short-circuit.

theta[s] = 'arccos(R[Ts]/Z[Ts])' ; theta[s] := evalf(arccos(R[Ts]/Z[Ts])*180/Pi) : 'theta[s]' = evalf(theta[s], 3) * `\260` ;N.B. The impedance angle of the secondary circuit is, as expected, equal to the absolute value of the angular displacement of the secondary current with respect to the secondary emf, which was computed earlier, i.e.,abs('theta[deg][i_s] - theta[deg][e_s]') = abs(evalf(delta[3], 3)) * `\260` ;

Z[b] := evalf(sqrt(R[b]^2 + (omega*L[b])^2)) : 'Z[b]' = evalf(Z[b]*10^3, 3)*m*Omega ;

theta[b] = 'arccos(R[b]/Z[b])' ; theta[b] := evalf(arccos(R[b]/Z[b])*180/Pi) : 'theta[b]' = evalf(theta[b], 3) * `\260` ;N.B. The impedance angle of the burden is, as expected, equal to the absolute value of the angular displacement of the secondary current with respect to the secondary voltage, which was computed earlier, i.e.,abs('theta[deg][i_s] - theta[deg][v_s]') = abs(evalf(delta[5], 3)) * `\260` ;

PF[s] = 'cos(R[Ts]/Z[Ts])' ; PF[s] := cos(R[Ts]/Z[Ts]) : 'PF[s]' = evalf(PF[s], 2) ;

PF[b] = 'cos(R[b]/Z[b])' ; PF[b] := cos(R[b]/Z[b]) : 'PF[b]' = evalf(PF[b], 2) ;

P[s] = 'I_s^2 * R[Ts]' ; P[s] := I_s^2 * R[Ts] : 'P[s]' = evalf(P[s], 3) * W ;

S[s] = 'I_s^2 * Z[Ts]' ; S[s] := I_s^2 * Z[Ts] : 'S[s]' = evalf(S[s], 3) * `V\267A` ;

The symbolic forms of the formulae for the functions obtained earlier in the angular domain may be required for some further analyses. These forms are made available at this point of the computational algorithm through a procedure, which is provided hereunder in steps.

theta[p_v] := theta[p] : I_p_mx_v := I_p_mx : L[b_v] := L[b] : L[Ts_v] := L[Ts] : N[p_v] := N[p] : N[s_v] := N[s] : omega_v := omega : R[m_v] := R[m] : R[b_v] := R[b] : R[Ts_v] := R[Ts] :

theta[p] := 'theta[p]' : I_p_mx := 'I_p_mx' : L[b] := 'L[b]' : L[Ts] := 'L[Ts]' : N[p] := 'N[p]' : N[s] := 'N[s]' : omega := 'omega' : R[m] := 'R[m]' : R[b] := 'R[b]' : R[Ts] := 'R[Ts]' :

(a) primary currenti[p](x) := i_p_f : 'i[p](x)' = i[p](x) ;(b) magnetizing current flowing in the primaryi[mp](x) := i_mp_f : 'i[mp](x)' = i[mp](x) ;(c) magnetic fluxphi[m](x) := phi_m_f : 'phi[m](x)' = phi[m](x) ;(d) secondary emfe[s](x) := e_s_f : 'e[s](x)' = e[s](x) ;(e) secondary currenti[s](x) := i_s_f : 'i[s](x)' = i[s](x) ;(f) primary emfe[p](x) := e_p_f : 'e[p](x)' = e[p](x) ;(g) secondary voltagev[s](x) := v_s_f : 'v[s](x)' = v[s](x) ;

All the formulae describing the time-varying parameters are again available in their symbolic forms and may be used for a further analysis, if required.

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